Vertical and horizontal

In physics, engineering and construction, the direction designated as vertical is usually that along which a plumb-bob hangs.

Alternatively, a spirit level that exploits the buoyancy of an air bubble and its tendency to go vertically upwards may be used to test for horizontality.

On the surface of a smoothly spherical, homogenous, non-rotating planet, the plumb bob picks out as vertical the radial direction.

This fact has real practical applications in construction and civil engineering, e.g., the tops of the towers of a suspension bridge are further apart than at the bottom.

It is a non homogeneous, non spherical, knobby planet in motion, and the vertical not only need not lie along a radial, it may even be curved and be varying with time.

[9][10] Neglecting the curvature of the earth, horizontal and vertical motions of a projectile moving under gravity are independent of each other.

For example, even a projectile fired in a horizontal direction (i.e., with a zero vertical component) may leave the surface of the spherical Earth and indeed escape altogether.

[13] In the context of a 1-dimensional orthogonal Cartesian coordinate system on a Euclidean plane, to say that a line is horizontal or vertical, an initial designation has to be made.

There is no special reason to choose the horizontal over the vertical as the initial designation: the two directions are on par in this respect.

This dichotomy between the apparent simplicity of a concept and an actual complexity of defining (and measuring) it in scientific terms arises from the fact that the typical linear scales and dimensions of relevance in daily life are 3 orders of magnitude (or more) smaller than the size of the Earth.

This is purely conventional (although it is somehow 'natural' when drawing a natural scene as it is seen in reality), and may lead to misunderstandings or misconceptions, especially in an educational context.

Spirit level bubble on a marble shelf tests for horizontality
A plumb bob
Strictly, vertical directions are never parallel on the surface of a spherical planet (except at opposite poles, where they are anti-parallel ).
On a spherical planet, horizontal planes intersect. In the example shown, the blue line represents the horizontal tangent plane at the North pole, the red the horizontal tangent plane at an equatorial point. The two intersect at a right angle.
In two dimensions. 1. The vertical direction is designated. 2. The horizontal is perpendicular to the vertical. Through any point P, there is exactly one vertical and exactly one horizontal. Alternatively, one can start by designating the horizontal direction.
The y -axis on the wall is vertical, but the one on the table is horizontal.
A spirit level on a shelf
Verticals at two separate points are not parallel. The same holds for their associated horizontal planes.
Field lines for a non-homogeneous knobbly planet in motion may be curved. The white, red and blue bits illustrate the planet's heterogeneity.
The y-axis on the wall is vertical, but the one on the table is horizontal.